Title: Finite Control Volume Analysis
1Finite Control Volume Analysis
Application of Reynolds Transport Theorem
2Moving from a System to a Finite Control Volume
- Mass
- Linear Momentum
- Moment of Momentum
- Energy
- Putting it all together!
3Conservation of Mass
B Total amount of ____ in the system b ____
per unit mass __
mass
1
mass
cv equation
But DMsys/Dt 0!
Continuity Equation
mass leaving - mass entering - rate of increase
of mass in cv
4Conservation of Mass
If mass in cv is constant
2
1
V1
A1
Unit vector is ______ to surface and pointed
____ of cv
normal
M/T
out
r
We assumed uniform ___ on the control surface
5Continuity Equation for Constant Density and
Uniform Velocity
Density is constant across cs
L3/T
Density is the same at cs1 and cs2
Simple version of the continuity equation for
conditions of constant density. It is understood
that the velocities are either ________ or
_______ ________.
uniform
spatially averaged
6Example Conservation of Mass?
The flow out of a reservoir is 2 L/s. The
reservoir surface is 5 m x 5 m. How fast is the
reservoir surface dropping?
h
Constant density
Velocity of the reservoir surface
Example
7Linear Momentum Equation
R.T.T.
momentum/unit mass
momentum
Steady state
8Linear Momentum Equation
Assumptions
- V fluid velocity relative to cv
Vectors!!!
9Steady Control Volume Form of Newtons Second Law
- What are the forces acting on the fluid in the
control volume?
- Shear forces at the walls
- Pressure forces at the walls
- Pressure forces on the ends
Why no shear on control surfaces?
_______________________________
No velocity gradient normal to surface
10Resultant Force on the Solid Surfaces
- The shear forces on the walls and the pressure
forces on the walls are generally the unknowns - Often the problem is to calculate the total force
exerted by the fluid on the solid surfaces - The magnitude and direction of the force
determines - size of _____________needed to keep pipe in place
- force on the vane of a pump or turbine...
thrust blocks
force applied by solid surfaces
11Linear Momentum Equation
Fp2
M2
Fssx
The momentum vectors have the same direction as
the velocity vectors
M1
Fssy
Fp1
W
12Example Reducing Elbow
2
Reducing elbow in vertical plane with water flow
of 300 L/s. The volume of water in the elbow is
200 L. Energy loss is negligible. Calculate the
force of the elbow on the fluid. W
_________ section 1 section 2 D 50 cm 30
cm A _________ _________ V _________ _________ p
150 kPa _________ M _________ _________ Fp ______
___ _________
1 m
1
-1961 N ?
0.196 m2
0.071 m2
y
1.53 m/s ?
4.23 m/s ?
?
-459 N ?
1269 N ?
Direction of V vectors
x
?
29,400 N ?
13Example What is p2?
Fp2 -9400 N
P2 132 kPa
14Example Reducing ElbowHorizontal Forces
2
Fp2
M2
1
y
Force of pipe on fluid
x
Pipe wants to move to the _________
left
15Example Reducing ElbowVertical Forces
2
W
1
Fp1
M1
y
28 kN acting downward on fluid
up
Pipe wants to move _________
x
16Example Fire nozzle
- A small fire nozzle is used to create a powerful
jet to reach far into a blaze. Estimate the force
that the water exerts on the fire nozzle. The
pressure at section 1 is 1000 kPa (gage). Ignore
frictional losses in the nozzle.
8 cm
2.5 cm
17Example Momentum with Complex Geometry
Find Q2, Q3 and force on the wedge.
?2
cs2
cs1
cs3
?1
?3
Q2, Q3, V2, V3, Fx
Unknown ________________
185 Unknowns Need 5 Equations
Unknowns Q2, Q3, V2, V3, Fx
Identify the 5 equations!
Continuity
Bernoulli (2x)
?2
cs2
cs1
Momentum (in x and y)
?1
cs3
?3
19Solve for Q2 and Q3
atmospheric pressure
Component of velocity in y direction
Mass conservation
Negligible losses apply Bernoulli
20Solve for Q2 and Q3
Why is Q2 greater than Q3?
21Solve for Fx
Force of wedge on fluid
22Vector solution
23Vector Addition
?2
cs2
cs1
cs3
?1
?3
Where is the line of action of Fss?
24Moment of Momentum Equation
R.T.T.
Moment of momentum
Moment of momentum/unit mass
Steady state
25Application to Turbomachinery
Vt
Vn
cs1
cs2
26Example Sprinkler
cs2
vt
?
?
10 cm
Total flow is 1 L/s. Jet diameter is 0.5
cm. Friction exerts a torque of 0.1 N-m-s2 ?2. ?
30º. Find the speed of rotation.
Vt and Vn are defined relative to control
surfaces.
27Example Sprinkler
a 0.1Nms2
b (1000 kg/m3)(0.001 m3/s) (0.1 m) 2 0.01 Nms
c -(1000 kg/m3)(0.001 m3/s)2(0.1m)(2sin30)/3.14/
(0.005 m)2
c -1.27 Nm
? 3.5/s
34 rpm
28Reflections
- What is the name of the equation that we used to
move from a system (Lagrangian) view to the
control volume (Eulerian) view? - Explain the analogy to your checking account.
- The velocities in the linear momentum equation
are relative to ? - Why is ma non-zero for a fixed control volume?
- Under what conditions could you generate power
from a rotating sprinkler? - What questions do you have about application of
the linear momentum and momentum of momentum
equations?
29Energy Equation
RTT
DE/Dt
What is for a system?
First law of thermodynamics The heat QH added to
a system plus the work W done on the system
equals the change in total energy E of the system.
30dE/dt for our System?
Pressure work
Shaft work
Heat transfer
31General Energy Equation
RTT
1st Law of Thermo
Potential
Kinetic
Internal (molecular spacing and forces)
Total
32Simplify the Energy Equation
0
Steady
Assume...
- Hydrostatic pressure distribution at cs
not uniform
But V is often ____________ over control surface!
33Energy Equation Kinetic Energy Term
V point velocity
If V normal to n
kinetic energy correction term
? _________________________
? ___ for uniform velocity
1
34Energy Equation steady, one-dimensional,
constant density
mass flux rate
35Energy Equation steady, one-dimensional,
constant density
divide by g
thermal
mechanical
Lost mechanical energy
hP
36Thermal Components of the Energy Equation
For incompressible liquids
Water specific heat 4184 J/(kgK)
Change in temperature
Heat transferred to fluid
Example
37Example Energy Equation(energy loss)
An irrigation pump lifts 50 L/s of water from a
reservoir and discharges it into a farmers
irrigation channel. The pump supplies a total
head of 10 m. How much mechanical energy is lost?
cs2
4 m
2 m
cs1
datum
38Example Energy Equation(pressure at pump outlet)
The total pipe length is 50 m and is 20 cm in
diameter. The pipe length to the pump is 12 m.
What is the pressure in the pipe at the pump
outlet? You may assume (for now) that the only
losses are frictional losses in the pipeline.
50 L/s
hP 10 m
4 m
cs2
2 m
cs1
datum
0 /
0 /
0 /
0 /
?
We need _______ in the pipe, __, and ____ ____.
velocity
head loss
39Example Energy Equation (pressure at pump
outlet)
- How do we get the velocity in the pipe?
- How do we get the frictional losses?
- What about a?
Q VA
A ?d2/4
V 4Q/(?d2)
V 4(0.05 m3/s)/? ?0.2 m)2 1.6 m/s
Expect losses to be proportional to length of the
pipe
hl (6 m)(12 m)/(50 m) 1.44 m
40Kinetic Energy Correction Term a
- a is a function of the velocity distribution in
the pipe. - For a uniform velocity distribution ____
- For laminar flow ______
- For turbulent flow _____________
- Often neglected in calculations because it is so
close to 1
a is 1
a is 2
1.01 lt a lt 1.10
41Example Energy Equation (pressure at pump
outlet)
V 1.6 m/s ? 1.05 hL 1.44 m
50 L/s
hP 10 m
4 m
2 m
datum
59.1 kPa
42Example Energy Equation(Hydraulic Grade Line -
HGL)
- We would like to know if there are any places in
the pipeline where the pressure is too high
(_________) or too low (water might boil -
cavitation). - Plot the pressure as piezometric head (height
water would rise to in a manometer) - How?
pipe burst
43Example Energy Equation(Hydraulic Grade Line -
HGL)
50 L/s
HP 10 m
4 m
2 m
datum
44EGL (or TEL) and HGL
Hydraulic Grade Line
Energy Grade Line
Piezometric head
Elevation head (w.r.t. datum)
velocity head
Pressure head (w.r.t. reference pressure)
45EGL (or TEL) and HGL
- The energy grade line may never be horizontal or
slope upward (in direction of flow) unless energy
is added (______) - The decrease in total energy represents the head
loss or energy dissipation per unit weight - EGL and HGL are ____________and lie at the free
surface for water at rest (reservoir) - Whenever the HGL falls below the point in the
system for which it is plotted, the local
pressures are lower than the __________________
pump
coincident
reference pressure
46Example HGL and EGL
velocity head
pressure head
energy grade line
hydraulic grade line
elevation
z
pump
z 0
datum
47Bernoulli vs. Control Volume Conservation of
Energy
Find the velocity and flow. How would you solve
these two problems?
pipe
Free jet
48Bernoulli vs. Control Volume Conservation of
Energy
Control surface to control surface
Point to point along streamline
Has a term for frictional losses
No frictional losses
Based on average velocity
Requires kinetic energy correction factor
Includes shaft work
49Power and Efficiencies
- Electrical power
- Shaft power
- Impeller power
- Fluid power
Motor losses
IE
bearing losses
Tshaft
pump losses
Tloss
?QHp
50Example Hydroplant
Water power ? total losses ? efficiency of
turbine ? efficiency of generator ?
50 m
2100 kW
Q 5 m3/s
116 kNm
180 rpm
solution
51Energy Equation Review
- Control Volume equation
- Simplifications
- steady
- constant density
- hydrostatic pressure distribution across control
surface (cs normal to streamlines) - Direction of flow matters (in vs. out)
- We dont know how to predict head loss
52Conservation of Energy, Momentum, and Mass
- Most problems in fluids require the use of more
than one conservation law to obtain a solution - Often a simplifying assumption is required to
obtain a solution - neglect energy losses (_______) over a short
distance with no flow expansion - neglect shear forces on the solid surface over a
short distance
to heat
53Head Loss Minor Losses
- Head (or energy) loss due tooutlets, inlets,
bends, elbows, valves, pipe size changes - Losses due to expansions are ________ than losses
due to contractions - Losses can be minimized by gradual transitions
- Losses are expressed in the formwhere K is the
loss coefficient
greater
54Head Loss due to Sudden ExpansionConservation
of Energy
1
2
zin zout
What is pin - pout?
55Head Loss due to Sudden ExpansionConservation
of Momentum
A2
A1
x
1
2
Apply in direction of flow
Neglect surface shear
Pressure is applied over all of section
1. Momentum is transferred over area
corresponding to upstream pipe diameter. Vin is
velocity upstream.
Divide by (Aout g)
56Head Loss due to Sudden Expansion
Energy
Mass
Momentum
Discharge into a reservoir?_________
KL1
57Example Losses due to Sudden Expansion in a Pipe
(Teams!)
- A flow expansion discharges 2.4 L/s directly into
the air. Calculate the pressure immediately
upstream from the expansion
1 cm
3 cm
We can solve this using either the momentum
equation or the energy equation (with the
appropriate term for the energy losses)!
Solution
58Summary
- Control volumes should be drawn so that the
surfaces are either tangent (no flow) or normal
(flow) to streamlines. - In order to solve a problem the flow surfaces
need to be at locations where all but 1 or 2 of
the energy terms are known - When possible choose a frame of reference so the
flows are steady
59Summary
- Control volume equation Required to make the
switch from Lagrangian to Eulerian - Any conservative property can be evaluated using
the control volume equation - mass, energy, momentum, concentrations of species
- Many problems require the use of several
conservation laws to obtain a solution
end
60Example Conservation of Mass(Team Work)
- The flow through the orifice is a function of the
depth of water in the reservoir - Find the time for the reservoir level to drop
from 10 cm to 5 cm. The reservoir surface is 15
cm x 15 cm. The orifice is 2 mm in diameter and
is 2 cm off the bottom of the reservoir. The
orifice coefficient is 0.6. - CV with constant or changing mass.
- Draw CV, label CS, solve using variables starting
with
61Example Conservation of MassConstant Volume
h
cs1
cs2
62Example Conservation of MassChanging Volume
h
cs1
cs2
63Example Conservation of Mass
64Pump Head
hp
65Example Venturi
66Example Venturi
Find the flow (Q) given the pressure drop between
section 1 and 2 and the diameters of the two
sections. Draw an appropriate control volume. You
may assume the head loss is negligible. Draw the
EGL and the HGL.
?h
2
1
67Example Venturi
68Fire nozzle Team Work
Identify what you need to know Count your
unknowns Determine what equations you will use
8 cm
2.5 cm
1000 kPa
69Find the Velocities
70Fire nozzle Solution
2.5 cm
8 cm
1000 kPa
Which direction does the nozzle want to go? ______
Is this the force that the firefighters need to
brace against? _______
NO!
force applied by nozzle on water
71Temperature Rise over Taughanock Falls
- Drop of 50 meters
- Find the temperature rise
72Hydropower
73Solution Losses due to Sudden Expansion in a Pipe
- A flow expansion discharges 2.4 L/s directly into
the air. Calculate the pressure immediately
upstream from the expansion
1 cm
3 cm